Opuscula Math. 31, no. 3 (2011), 411-424

Opuscula Mathematica

Existence and asymptotic behavior of solutions for Hénon type equations

Wei Long
Jianfu Yang

Abstract. This paper is concerned with ground state solutions for the Hénon type equation \(-\Delta u(x)=|y|^{\alpha} u^{p-1}(x)\) in \(\Omega\), where \(\Omega=B^k(0,1)\times B^{n-k}(0,1)\subset \mathbb{R}^n\) and \(x=(y,z) \in \mathbb{R}^k \times \mathbb{R}^{n-k}\). We study the existence of cylindrically symmetric and non-cylindrically symmetric ground state solutions for the problem. We also investigate asymptotic behavior of the ground state solution when \(p\) tends to the critical exponent \(2^*=\frac {2n}{n-2}\) if \(n\geq 3\).

Keywords: Hénon equation, cylindrical symmetry, non-cylindrical symmetry, asymptotic behavior.

Mathematics Subject Classification: 35J50, 35J55, 35J60.

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Cite this article as:
Wei Long, Jianfu Yang, Existence and asymptotic behavior of solutions for Hénon type equations, Opuscula Math. 31, no. 3 (2011), 411-424, http://dx.doi.org/10.7494/OpMath.2011.31.3.411

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